108 research outputs found
ΠΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΏΡΠΈΠΌΠΈΡΠΈΠ²Π½ΡΡ ΡΡ Π΅ΠΌ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Ρ ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ°ΠΌΠΈ
The paper considers algebraic program models with procedures designed to analyze program semantic properties based on program schemes. This leads to the problem of program scheme equivalence and the problem of constructing a complete system of equivalent program scheme transformations. Among algebraic program models with procedures, we focus on gateway models induced by program models without procedures and primitive program schemes belonging to these gateway models. The equivalence problem is decidable for these schemes. In the case where the inducer is a special type of program model without procedures, we construct a complete system of equivalent scheme transformations for a particular subclass of primitive program schemes.Π ΡΡΠ°ΡΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°ΡΡΡΡ Π°Π»Π³Π΅Π±ΡΠ°ΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Ρ ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ°ΠΌΠΈ, ΠΏΡΠ΅Π΄Π½Π°Π·Π½Π°ΡΠ΅Π½Π½ΡΠ΅ Π΄Π»Ρ ΠΈΠ·ΡΡΠ΅Π½ΠΈΡ ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ²ΠΎΠΉΡΡΠ² ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Π½Π° ΠΈΡ
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Π΅ΠΌΠ°Ρ
. Π’Π°ΠΊ Π²ΠΎΠ·Π½ΠΈΠΊΠ°ΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ ΡΡ
Π΅ΠΌ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ ΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΠΏΠΎΠ»Π½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΡΡ
ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΡΡ
Π΅ΠΌ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ. Π‘ΡΠ΅Π΄ΠΈ Π°Π»Π³Π΅Π±ΡΠ°ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Ρ ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ°ΠΌΠΈ Π²ΡΠ΄Π΅Π»Π΅Π½Ρ ΠΏΠ΅ΡΠ΅Π³ΠΎΡΠΎΠ΄ΡΠ°ΡΡΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈ, ΠΈΠ½Π΄ΡΡΠΈΡΡΠ΅ΠΌΡΠ΅ ΠΌΠΎΠ΄Π΅Π»ΡΠΌΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Π±Π΅Π· ΠΏΡΠΎΡΠ΅Π΄ΡΡ, ΠΈ ΠΏΡΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°ΡΠΈΠ΅ ΠΈΠΌ ΠΏΡΠΈΠΌΠΈΡΠΈΠ²Π½ΡΠ΅ ΡΡ
Π΅ΠΌΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ. ΠΠ»Ρ Π½ΠΈΡ
ΡΠ°Π·ΡΠ΅ΡΠΈΠΌΠ° ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΡΠ°ΡΡΠ΅ Π² ΡΠ»ΡΡΠ°Π΅, ΠΊΠΎΠ³Π΄Π° ΠΈΠ½Π΄ΡΡΠΈΡΡΡΡΠ΅ΠΉ ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΡΠ°Π²Π½ΠΎΠ²Π΅ΡΠ΅Π½Π½Π°Ρ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠΎΠ²Π°Ρ ΠΌΠΎΠ΄Π΅Π»Ρ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Ρ Π»Π΅Π²ΡΠΌ ΡΠΎΠΊΡΠ°ΡΠ΅Π½ΠΈΠ΅ΠΌ, Π΄Π»Ρ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΠΎΠ³ΠΎ ΠΏΠΎΠ΄ΠΊΠ»Π°ΡΡΠ° ΠΏΡΠΈΠΌΠΈΡΠΈΠ²Π½ΡΡ
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Π΅ΠΌ ΠΏΠΎΡΡΡΠΎΠ΅Π½Π° ΠΏΠΎΠ»Π½Π°Ρ Π² Π½Π΅ΠΌ ΡΠΈΡΡΠ΅ΠΌΠ° ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΡΡ
ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΠΉ ΡΡ
Π΅ΠΌ
Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes
In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis.
The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares.
From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations.
The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.Siirretty Doriast
Searching the space of representations: reasoning through transformations for mathematical problem solving
The role of representation in reasoning has been long and widely regarded as crucial.
It has remained one of the fundamental considerations in the design of information-processing
systems and, in particular, for computer systems that reason. However, the
process of change and choice of representation has struggled to achieve a status as a
task for the systems themselves. Instead, it has mostly remained a responsibility for
the human designers and programmers.
Many mathematical problems have the characteristic of being easy to solve only
after a unique choice of representation has been made. In this thesis we examine two
classes of problems in discrete mathematics which follow this pattern, in the light of
automated and interactive mechanical theorem provers. We present a general notion of
structural transformation, which accounts for the changes of representation seen in such
problems, and link this notion to the existing Transfer mechanism in the interactive
theorem prover Isabelle/HOL.
We present our mechanisation in Isabelle/HOL of some specific transformations identified as key in the solutions of the aforementioned mathematical problems. Furthermore,
we present some tools that we developed to extend the functionalities of the
Transfer mechanism, designed with the specific purpose of searching efficiently the
space of representations using our set of transformations. We describe some experiments
that we carried out using these tools, and analyse these results in terms of how
close the tools lead us to a solution, and how desirable these solutions are.
The thorough qualitative analysis we present in this thesis reveals some promise as
well as some challenges for the far-reaching problem of representation in reasoning, and
the automation of the processes of change and choice of representation
Advanced Concepts in Particle and Field Theory
Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication
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